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       "\t    keyboard_manager: IPython.keyboard_manager\n",
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       "</script>\n",
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     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "using PotentialFlow\n",
    "using Plots\n",
    "\n",
    "using Interact\n",
    "\n",
    "clibrary(:colorbrewer);\n",
    "streamline_colors = cgrad([:black, :black]);\n",
    "default(ratio = 1, legend = false);"
   ]
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  {
   "cell_type": "markdown",
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   "source": [
    "# The Doublet"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "Given a freestream in the $x$-direction, we want to place potential flow elements to form a cylinder centered at the origin."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": false,
    "slideshow": {
     "slide_type": "-"
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   },
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    "freestream = Freestream(1.0);\n",
    "ψ_levels = collect(linspace(-5, 5, 31))\n",
    "X = Y = linspace(-3.5, 3.5, 100)\n",
    "streamlines(X, Y, freestream, levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)"
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   "source": [
    "Let's try to place a point vortex at the origin to redirect flow over/under the cylinder."
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   "execution_count": 3,
   "metadata": {
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    }
   ],
   "source": [
    "@manipulate for dir in [:cw, :ccw]\n",
    "    point = Vortex.Point(0.0, dir == :cw ? -2π : 2π)\n",
    "    streamlines(X, Y, ([point], freestream), levels = ψ_levels, color = streamline_colors)\n",
    "    plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   },
   "source": [
    "So we want a CW vortex above the origin and a CCW vortex below the origin:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "-"
    }
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    "@manipulate for Δy in linspace(0, 1, 10)\n",
    "point = Vortex.Point.(0.5Δy*[im, -im], [-2π, 2π])\n",
    "streamlines(X, Y, (point, freestream), levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)\n",
    "end"
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   "cell_type": "markdown",
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   "source": [
    "We see that as we move the vortices closer, the closed streamline becomese more circular.\n",
    "However, its radius becomes smaller."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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     "slide_type": "-"
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     "execution_count": 5,
     "metadata": {
      "comm_id": "d372a8b7-7981-4880-a02d-e3a4faa5b19e",
      "reactive": true
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   "source": [
    "@manipulate for Δy = slider(linspace(0, 2, 10), value = 1.0)\n",
    "points = Vortex.Point.(0.5Δy*[im, -im], [-2π, 2π]./Δy)\n",
    "streamlines(X, Y, (points, freestream), levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can scale their circulations to be inversely proportional to their distance.\n",
    "Taking the limit as their distance go to zero, we get a **doublet**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
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     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "doublet = PotentialFlow.Doublets.Doublet(0.0im, π)\n",
    "streamlines(X, Y, [freestream, doublet], levels = ψ_levels, color = streamline_colors)"
   ]
  },
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     "slide_type": "slide"
    }
   },
   "source": [
    "# Method of Images"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the previous discussion, we talked about how we can create flow obstacles using source/vortex sheets.\n",
    "Another way to deal with solid surfaces in potential flow is to take advantage of certain flow symmetries."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Two point vortices with opposite signed circulation placed across the $x$-axis will eliminate the vertical velocities along the $x$-axis"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "points = Vortex.Point.([im, -im], [1, -1])\n",
    "streamlines(X, Y, points, color = streamline_colors)"
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  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "## How would we set up a system of images to find the streamlines of a point vortex near a corner?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
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      "comm_id": "324b1a7d-99a6-4b2e-a03c-6ae5401ababd",
      "reactive": true
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   "source": [
    "Γ₁ = 1.0\n",
    "zs = [1 + im, -1 + im, -1 - im, 1 - im]\n",
    "@manipulate for Γ₂ in linspace(-Γ₁, Γ₁, 51),\n",
    "                Γ₃ in linspace(-Γ₁, Γ₁, 51),\n",
    "                Γ₄ in linspace(-Γ₁, Γ₁, 51)\n",
    "    points = Vortex.Point.(zs, [Γ₁, Γ₂, Γ₃, Γ₄])\n",
    "    streamlines(X, Y, points, color = streamline_colors)\n",
    "    plot!(points, markersize = 10, color = :RdBu)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let's go back to the simple 2 point vortex case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "points = Vortex.Point.([im, -im], [1, -1])\n",
    "streamlines(X, Y, points, color = streamline_colors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Now let's move these pair of vortices so that we shift and rotate our wall"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point = Vortex.Point(-0.7 + 1.2im, 2π)\n",
    "image = Vortex.Point(conj(1/point.z), -2π)\n",
    "streamlines(X, Y, (point, image), levels = ψ_levels, color = streamline_colors)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Notice how the contours form closed circles:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "point = Vortex.Point(-0.7 + 1.2im, 2π)\n",
    "image = Vortex.Point(conj(1/point.z), -2π)\n",
    "streamlines(X, Y, (point, image), levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Since these streamlines are just contours showing curves with equal streamfunction values, the marked cylinder can be preserved if we superimpose another potential flow field that also has the cylinder as a closed streamline."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point = Vortex.Point(-0.7 + 1.2im, 2π)\n",
    "image = Vortex.Point(conj(1/point.z), -2π)\n",
    "images = streamlines(X, Y, (point, image), levels = ψ_levels, color = streamline_colors)\n",
    "cylinder = streamlines(X, Y, [freestream, doublet], levels = ψ_levels, color = streamline_colors)\n",
    "plot(images, cylinder, layout = (1, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
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      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "point = Vortex.Point(-0.7 + 1.2im, 2π)\n",
    "image = Vortex.Point(conj(1/point.z), -2π)\n",
    "streamlines(X, Y, (freestream, PotentialFlow.Doublets.Doublet(0.0im, π), point, image), \n",
    "    levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "Let's consider another approach to the same result.\n",
    "Imagine if we had a perfectly good cylinder flow.\n",
    "Then we deicde to put a vortex somewhere outside."
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   "source": [
    "point = Vortex.Point(-0.7 + 1.2im, 2π)\n",
    "streamlines(X, Y, (freestream, PotentialFlow.Doublets.Doublet(0.0im, π), point), levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)"
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   "source": [
    "We now want to place an image vortex that would reform the wall of the cylinder:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 15,
     "metadata": {
      "comm_id": "542f6dd1-3316-4ee1-8e36-201f08ae4ece",
      "reactive": true
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for x in linspace(-2, 2, 21)\n",
    "point = Vortex.Point(x + 1.2im, 10)\n",
    "image = Vortex.Point(conj(1/point.z), -10)\n",
    "streamlines(X, Y, (freestream, PotentialFlow.Doublets.Doublet(0.0im, π), point, image), levels = ψ_levels, color = streamline_colors)\n",
    "plot!(cos.(0:0.1:2π), sin.(0:0.1:2π), linestyle = :dash)\n",
    "end"
   ]
  }
 ],
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